sums of powers

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1: 24.20 Tables

§24.20 Tables

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2: 24.4 Basic Properties

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§24.4(iii) Sums of Powers

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3: 27.2 Functions

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27.2.6 $${\varphi }_k\left(n\right)=\sum _{\left(m,n\right)=1}{m}^k,$$

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Defines:

${\varphi }_k\left(n\right)$: sum of powers of integers relatively prime to a number

Symbols:

$\left(m,n\right)$: greatest common divisor (gcd), $k$: positive integer, $m$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.2.E6

Encodings:

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See also:

Annotations for §27.2(i), §27.2 and Ch.27

►the sum of the $k$th powers of the positive integers $m\le n$ that are relatively prime to $n$. ►

27.2.7 $$\varphi \left(n\right)={\varphi }_0\left(n\right).$$

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Defines:

$\varphi \left(n\right)$: Euler’s totient

Symbols:

${\varphi }_k\left(n\right)$: sum of powers of integers relatively prime to a number and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.2.E7

Encodings:

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Annotations for §27.2(i), §27.2 and Ch.27

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27.2.10 $${\sigma }_{\alpha }\left(n\right)=\sum _{d|n}{d}^{\alpha },$$

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Defines:

${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number

Symbols:

$d$: positive integer, $n$: positive integer and $\alpha$: parameter

A&S Ref:

24.3.3 I.A

Referenced by:

§27.14(ii)

Permalink:

http://dlmf.nist.gov/27.2.E10

Encodings:

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Annotations for §27.2(i), §27.2 and Ch.27

►is the sum of the $\alpha$th powers of the divisors of $n$, where the exponent $\alpha$ can be real or complex. …

4: 27.3 Multiplicative Properties

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27.3.6 $${\sigma }_{\alpha }\left(n\right)=\prod _{r=1}^{\nu \left(n\right)}\frac{p_r^{\alpha (1+a_r)}-1}{p_r^{\alpha }-1},$$ $\alpha \ne 0$.

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Symbols:

${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers and $a_r$: positive exponent

A&S Ref:

24.3.3 I.C

Permalink:

http://dlmf.nist.gov/27.3.E6

Encodings:

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Annotations for §27.3 and Ch.27

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27.3.7 $${\sigma }_{\alpha }\left(m\right){\sigma }_{\alpha }\left(n\right)=\sum _{d|\left(m,n\right)}{d}^{\alpha }{\sigma }_{\alpha }\left(\frac{mn}{d^2}\right),$$

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Symbols:

${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $\left(m,n\right)$: greatest common divisor (gcd), $d$: positive integer, $m$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.3.E7

Encodings:

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See also:

Annotations for §27.3 and Ch.27

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5: 27.7 Lambert Series as Generating Functions

… ►If , then the quotient $x^n/(1-x^n)$ is the sum of a geometric series, and when the series (27.7.1) converges absolutely it can be rearranged as a power series: … ►

27.7.5 $$\sum _{n=1}^{\mathrm{\infty }}{n}^{\alpha }\frac{x^n}{1-x^n}=\sum _{n=1}^{\mathrm{\infty }}{\sigma }_{\alpha }\left(n\right)x^n,$$

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Symbols:

${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $n$: positive integer and $x$: real number

A&S Ref:

24.3.3 I.B

Permalink:

http://dlmf.nist.gov/27.7.E5

Encodings:

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See also:

Annotations for §27.7 and Ch.27

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6: 27.6 Divisor Sums

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27.6.6 $$\sum _{d|n}{\varphi }_k\left(d\right){\left(\frac{n}{d}\right)}^k=1^k+2^k+\mathrm{\cdots }+n^k,$$

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Symbols:

${\varphi }_k\left(n\right)$: sum of powers of integers relatively prime to a number, $d$: positive integer, $k$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.6.E6

Encodings:

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See also:

Annotations for §27.6 and Ch.27

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7: 24.19 Methods of Computation

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Tanner and Wagstaff (1987) derives a congruence $(modp)$ for Bernoulli numbers in terms of sums of powers. See also §24.10(iii).

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8: Bibliography

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Bernoulli’s power-sum formulas revisited. Math. Gaz. 90 (518), pp. 276–279.

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Referenced by:

§24.4(iii).

Encodings:

BibTeX

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9: 27.4 Euler Products and Dirichlet Series

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27.4.11 $$\sum _{n=1}^{\mathrm{\infty }}{\sigma }_{\alpha }\left(n\right)n^{-s}=\zeta \left(s\right)\zeta \left(s-\alpha \right),$$ $\mathrm{\Re }s>\max (1,1+\mathrm{\Re }\alpha )$,

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, ${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $\mathrm{\Re }$: real part and $n$: positive integer

A&S Ref:

24.3.3 I.B

Permalink:

http://dlmf.nist.gov/27.4.E11

Encodings:

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See also:

Annotations for §27.4 and Ch.27

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10: 27.11 Asymptotic Formulas: Partial Sums

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27.11.4 $$\sum _{n\le x}{\sigma }_1\left(n\right)=\frac{{\pi }^2}{12}{x}^2+O\left(x\ln x\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, ${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $\ln z$: principal branch of logarithm function, $n$: positive integer and $x$: real number

A&S Ref:

24.3.3 III (in slightly different form)

Permalink:

http://dlmf.nist.gov/27.11.E4

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.11 and Ch.27

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27.11.5 $$\sum _{n\le x}{\sigma }_{\alpha }\left(n\right)=\frac{\zeta \left(\alpha +1\right)}{\alpha +1}{x}^{\alpha +1}+O\left(x^{\beta }\right),$$ $\alpha >0$, $\alpha \ne 1$, $\beta =\max (1,\alpha )$.

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Symbols:

$O\left(x\right)$: order not exceeding, $\zeta \left(s\right)$: Riemann zeta function, ${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $n$: positive integer and $x$: real number

Permalink:

http://dlmf.nist.gov/27.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §27.11 and Ch.27

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